Download 1 Definition of a Scale Parameter

Avd. Matematisk statistik
Sf 2955: Computer intensive methods :
SCALE PARAMETER/ Timo Koski
The notation
F (x; θ)
denotes a distribution function that depends on a parameter θ. For example,
1 − e−x/θ if x ≥ 0
F (x; θ) =
0
if x < 0,
is the distribution function of the exponential distribution Exp (θ).
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Definition of a Scale Parameter
We define a scale parameter.
Definition 1.1 Assume θ > 0 in F (x; θ). Then θ is a scale parameter, if it
holds for all x that
x
F (x; θ) = H
,
(1.1)
θ
where H(x) is a distribution function.
To take an example, θ is scale parameter in the exponential distribution
Exp (θ), as we have
x
F (x; θ) = H
,
θ
where
1 − e−x if x ≥ 0
H(x) =
0
if x < 0.
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Properties, Pivotal Variables
We observe two lemmas.
Lemma 2.1 Assume X ∈ F (x; θ). θ is a scale parameter if and only if the
distribution of
X
θ
does not depend on θ.
Proof: ⇒: Assume θ is a scale parameter. Then the distribution of
by
X
P
≤ z = P (X ≤ θz)
θ
X
θ
is given
since θ > 0 by definition. Then, as we assume that θ is a scale parameter,
θz
= H (z) ,
P (X ≤ θz) = F (θz; θ) = H
θ
or
P
X
≤z
θ
= H(z)
which says that Xθ has distribution which does not depend on θ.
⇐: We assume that Xθ has distribution, say H, which does not depend on
θ > 0. Then
X
x
F (x; θ) = P (X ≤ x; θ) = P
≤ ;θ
θ
θ
since θ > 0. But by assumption
x
X
x
P
≤ ;θ = H
θ
θ
θ
where the distribution function H (z) does not depend on θ. But then we
have for any x shown that
x
F (x; θ) = H
θ
which in view of (1.1) gives the claim as asserted.
This lemma says that Xθ is a pivotal variable.
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Remark 2.1 Note that a pivotal variable need not be a statistic − the variable can depend on parameters of the model, but its distribution must not.
If it is a statistic, then it is known as an ancillary statistic. Pivotal quantities
are used to the construction of test statistics, e.g., Student’s t−statistic is
pivotal for a normal distribution with unknown variance (and mean). Pivotal
variables provide in addition a method of constructing confidence intervals,
and the use of pivotal quantities improves performance of the bootstrap, as
defined later.
d
F (x; θ) = f (x; θ) for all x.
Lemma 2.2 Assume F (x; θ) has a density dx
Then θ is a scale parameter if and only if
1 x
f (x; θ) = g
,
(2.2)
θ
θ
where g is a probability density.
Proof: ⇒: If F (x; θ) has an integrable derivative and θ is a scale parameter,
then it holds from (1.1 ) that
d x 1 x
d
= g
F (x; θ) =
H
f (x; θ) =
dx
dx
θ
θ
θ
where we have set g(x) =
distribution function.
⇐: We assume that
d
H
dx
(x), which is a density function, as H is a
1 x
.
f (x; θ) = g
θ
θ
Z x
Z
1 x u
F (x; θ) =
f (u; θ)du =
du.
g
θ −∞
θ
−∞
Then
Here we make a change of variable t = uθ , du = θdt and thus
Z
Z x
x
θ
1 x u
g (t) dt = G
du =
.
g
θ −∞
θ
θ
−∞
where
d
G (x)
dx
= g(x). In other words we have obtained for every x that
x
,
F (x; θ) = G
θ
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and since G (x) is a distribution function, the desired assertion follows by
(1.1).
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Examples and the Scale-Free property
We give first some further examples.
1. X ∈ N (µσ, σ 2 ). We guess that σ is a scale parameter. The pertinent
density is
(x−µσ)2
1
e− 2σ2
f (x; σ) = √
2πσ
and some elementary algebra gives
f (x; σ) = √
x −µ)2
(σ
1
1 x
e− 2 = g
,
σ
σ
2πσ
where g is the density of N(µ, 1) or
(z−µ)2
1
g (z) = √ e− 2
2π
and the desired conclusion follows by (2.2).
2. X ∈ R (0, θ). Then we take x ∈ [0, 1] and get
θx
X
≤ x = P (X ≤ θx) =
=x
P
θ
θ
as the distribution function of R(0, θ) is F (z) = θz . The desired conclusion follows now from the first lemma. We have shown that Xθ ∈ R(0, 1),
which is, however, immediate from the definitions.
3. Finally, we take the Weibull distribution, so that
k
1 − e−(x/λ)
if x ≥ 0
F (x; λ) =
0
if x < 0.
For k = 1 gives the exponential distribution, and k = 2 gives the
Rayleigh distribution. It is obvious that λ > 0 is scale parameter.
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Suppose that X ∈ Exp (θ) is a lifetime measured in seconds. Then θ =
E [X] is also measured in seconds. We might convert seconds to minutes.
But the probability
x
P (X ≤ x) = H
,
θ
is the same whether we measure in minutes or seconds, i.e., it is invariant
with respect to scale or the units of measurement, as is quite reasonable.
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